Given the quadratic equation \( x^{2} + 6x + 9 = 0 \), determine the nature of its roots using the discriminant.

To determine the nature of the roots of the quadratic equation \( x^{2} + 6x + 9 = 0 \), we can use the discriminant formula \( D = b^2 - 4ac \). Here, \( a = 1 \), \( b = 6 \), and \( c = 9 \). Plugging in the values, we get: \[ D = 6^2 - 4(1)(9) = 36 - 36 = 0 \] Since the discriminant \( D \) is equal to 0, this indicates that the quadratic equation has exactly one real root (a double root), meaning the graph of the equation touches the x-axis at a single point. For a quirky fun fact, the equation can be factored as \( (x + 3)^2 = 0 \), which clearly demonstrates that the root is \( x = -3 \). It’s like finding a hidden treasure that just happens to sit nicely at the same spot! Quadratics with a zero discriminant often indicate scenarios in the real world where stability is achieved at a single point, such as an object perfectly balanced on a pivot. Isn’t it fascinating how math mirrors real-life situations?